--- trunk/multipole/multipole_2/multipole2.tex	2014/06/14 15:24:30	4183
+++ trunk/multipole/multipole_2/multipole2.tex	2014/06/14 23:35:39	4184
@@ -70,27 +70,27 @@ of Notre Dame, Notre Dame, IN 46556}
   We have tested the real-space shifted potential (SP),
   gradient-shifted force (GSF), and Taylor-shifted force (TSF) methods
   for multipoles that were developed in the first paper in this series
-  against a reference method. The tests were carried out in a variety
-  of condensed-phase environments which were designed to test all
-  levels of the multipole-multipole interactions.  Comparisons of the
-  energy differences between configurations, molecular forces, and
-  torques were used to analyze how well the real-space models perform
-  relative to the more computationally expensive Ewald sum.  We have
-  also investigated the energy conservation properties of the new
-  methods in molecular dynamics simulations using all of these
-  methods. The SP method shows excellent agreement with
-  configurational energy differences, forces, and torques, and would
-  be suitable for use in Monte Carlo calculations.  Of the two new
-  shifted-force methods, the GSF approach shows the best agreement
-  with Ewald-derived energies, forces, and torques and exhibits energy
-  conservation properties that make it an excellent choice for
-  efficiently computing electrostatic interactions in molecular
-  dynamics simulations.
+  against the multipolar Ewald sum as a reference method. The tests
+  were carried out in a variety of condensed-phase environments which
+  were designed to test all levels of the multipole-multipole
+  interactions.  Comparisons of the energy differences between
+  configurations, molecular forces, and torques were used to analyze
+  how well the real-space models perform relative to the more
+  computationally expensive Ewald treatment.  We have also investigated the
+  energy conservation properties of the new methods in molecular
+  dynamics simulations using all of these methods. The SP method shows
+  excellent agreement with configurational energy differences, forces,
+  and torques, and would be suitable for use in Monte Carlo
+  calculations.  Of the two new shifted-force methods, the GSF
+  approach shows the best agreement with Ewald-derived energies,
+  forces, and torques and exhibits energy conservation properties that
+  make it an excellent choice for efficiently computing electrostatic
+  interactions in molecular dynamics simulations.
 \end{abstract}
 
 %\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy
                              % Classification Scheme.
-\keywords{Electrostatics, Multipoles, Real-space}
+%\keywords{Electrostatics, Multipoles, Real-space}
 
 \maketitle
 
@@ -122,38 +122,39 @@ periodicity in the Ewald’s method can also be proble
 interfaces.\cite{Parry:1975if,Parry:1976fq,Clarke77,Perram79,Rhee:1989kl}
 To simulate interfacial systems, Parry's extension of the 3D Ewald sum
 is appropriate for slab geometries.\cite{Parry:1975if} The inherent
-periodicity in the Ewald’s method can also be problematic for
-interfacial molecular systems.\cite{Fennell:2006lq} Modified Ewald
-methods that were developed to handle two-dimensional (2D)
-electrostatic interactions in interfacial systems have not had similar
-particle-mesh treatments.\cite{Parry:1975if, Parry:1976fq, Clarke77,
-  Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq}
+periodicity in the Ewald method can also be problematic for molecular
+interfaces.\cite{Fennell:2006lq} Modified Ewald methods that were
+developed to handle two-dimensional (2D) electrostatic interactions in
+interfacial systems have not seen similar particle-mesh
+treatments,\cite{Parry:1975if, Parry:1976fq, Clarke77,
+  Perram79,Rhee:1989kl,Spohr:1997sf,Yeh:1999oq} and still scale poorly
+with system size.
 
 \subsection{Real-space methods}
 Wolf \textit{et al.}\cite{Wolf:1999dn} proposed a real space $O(N)$
 method for calculating electrostatic interactions between point
 charges. They argued that the effective Coulomb interaction in
-condensed systems is actually short ranged.\cite{Wolf92,Wolf95} For an
-ordered lattice (e.g., when computing the Madelung constant of an
-ionic solid), the material can be considered as a set of ions
+condensed phase systems is actually short ranged.\cite{Wolf92,Wolf95}
+For an ordered lattice (e.g., when computing the Madelung constant of
+an ionic solid), the material can be considered as a set of ions
 interacting with neutral dipolar or quadrupolar ``molecules'' giving
 an effective distance dependence for the electrostatic interactions of
 $r^{-5}$ (see figure \ref{fig:schematic}).  For this reason, careful
-applications of Wolf's method are able to obtain accurate estimates of
-Madelung constants using relatively short cutoff radii.  Recently,
-Fukuda used neutralization of the higher order moments for the
-calculation of the electrostatic interaction of the point charges
-system.\cite{Fukuda:2013sf}
+application of Wolf's method can obtain accurate estimates of Madelung
+constants using relatively short cutoff radii.  Recently, Fukuda used
+neutralization of the higher order moments for calculation of the
+electrostatic interactions in point charge
+systems.\cite{Fukuda:2013sf}
 
 \begin{figure}
   \centering
   \includegraphics[width=\linewidth]{schematic.pdf}
   \caption{Top: Ionic systems exhibit local clustering of dissimilar
     charges (in the smaller grey circle), so interactions are
-    effectively charge-multipole in order at longer distances.  With
-    hard cutoffs, motion of individual charges in and out of the
-    cutoff sphere can break the effective multipolar ordering.
-    Bottom: dipolar crystals and fluids have a similar effective
+    effectively charge-multipole at longer distances.  With hard
+    cutoffs, motion of individual charges in and out of the cutoff
+    sphere can break the effective multipolar ordering.  Bottom:
+    dipolar crystals and fluids have a similar effective
     \textit{quadrupolar} ordering (in the smaller grey circles), and
     orientational averaging helps to reduce the effective range of the
     interactions in the fluid.  Placement of reversed image multipoles
@@ -163,7 +164,7 @@ truncation defects. Wolf \textit{et al.} further argue
 \end{figure}
 
 The direct truncation of interactions at a cutoff radius creates
-truncation defects. Wolf \textit{et al.} further argued that
+truncation defects. Wolf \textit{et al.}  argued that
 truncation errors are due to net charge remaining inside the cutoff
 sphere.\cite{Wolf:1999dn} To neutralize this charge they proposed
 placing an image charge on the surface of the cutoff sphere for every
@@ -339,7 +340,7 @@ $\bf a$.
 where the multipole operator for site $\bf a$, $\hat{M}_{\bf a}$, is
 expressed in terms of the point charge, $C_{\bf a}$, dipole, ${\bf D}_{\bf
     a}$, and quadrupole, $\mathbf{Q}_{\bf a}$, for object
-$\bf a$.
+$\bf a$, etc.
 
 % Interactions between multipoles can be expressed as higher derivatives
 % of the bare Coulomb potential, so one way of ensuring that the forces
@@ -412,7 +413,7 @@ to another site within cutoff sphere are derived from
 connection to unmodified electrostatics as well as the smooth
 transition to zero in both these functions as $r\rightarrow r_c$.  The
 electrostatic forces and torques acting on the central multipole due
-to another site within cutoff sphere are derived from
+to another site within the cutoff sphere are derived from
 Eq.~\ref{generic}, accounting for the appropriate number of
 derivatives. Complete energy, force, and torque expressions are
 presented in the first paper in this series (Reference
@@ -430,7 +431,7 @@ U_{D_{\bf a} D_{\bf b}}(r_c)
 U_{D_{\bf a}D_{\bf b}}(r)  = U_{D_{\bf a}D_{\bf b}}(r) -
 U_{D_{\bf a} D_{\bf b}}(r_c)
    - (r_{ab}-r_c) ~~~\hat{r}_{ab} \cdot
-  \vec{\nabla} U_{D_{\bf a}D_{\bf b}}(r) \Big \lvert _{r_c}
+  \nabla U_{D_{\bf a}D_{\bf b}}(r_c).
 \end{equation}
 Here the lab-frame orientations of the two dipoles, $\mathbf{D}_{\bf
   a}$ and $\mathbf{D}_{\bf b}$, are retained at the cutoff distance
@@ -454,18 +455,21 @@ U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathb
 In general, the gradient shifted potential between a central multipole
 and any multipolar site inside the cutoff radius is given by,
 \begin{equation}
-U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) -
-U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{r}
-\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert  _{r_c} \right]
+  U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) -
+    U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{\mathbf{r}}
+    \cdot \nabla U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right]
 \label{generic2}
 \end{equation}
 where the sum describes a separate force-shifting that is applied to
-each orientational contribution to the energy. 
+each orientational contribution to the energy.  In this expression,
+$\hat{\mathbf{r}}$ is the unit vector connecting the two multipoles
+($a$ and $b$) in space, and $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$
+represent the orientations the multipoles.
 
 The third term converges more rapidly than the first two terms as a
 function of radius, hence the contribution of the third term is very
 small for large cutoff radii.  The force and torque derived from
-equation \ref{generic2} are consistent with the energy expression and
+Eq. \ref{generic2} are consistent with the energy expression and
 approach zero as $r \rightarrow r_c$.  Both the GSF and TSF methods
 can be considered generalizations of the original DSF method for
 higher order multipole interactions. GSF and TSF are also identical up
@@ -473,7 +477,7 @@ GSF potential are presented in the first paper in this
 the energy, force and torque for higher order multipole-multipole
 interactions. Complete energy, force, and torque expressions for the
 GSF potential are presented in the first paper in this series
-(Reference~\onlinecite{PaperI})
+(Reference~\onlinecite{PaperI}).
 
 
 \subsection{Shifted potential (SP) }
@@ -487,7 +491,7 @@ U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) \ri
 effectively shifts the total potential to zero at the cutoff radius,
 \begin{equation}
 U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) -
-U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right]
+U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right]
 \label{eq:SP}
 \end{equation}      	
 where the sum describes separate potential shifting that is done for
